ABSTRACT
Let (X, || • ||) be a Banach space and r a topology on X . If C is a subset of X , a map ping T : C -► C is said to be asymptotically regular if lim„ \\Tn+{x — r w*|| = 0 for every x e C [2]. The existence of fixed points for asymptotically regular mappings has been studied by several authors [3], [4], [7], [9], [11], [13], [14], [15], [19], [20]. When C is a convex set, it is known [16] that the problem of the existence of fixed points for a nonexpansive mapping T : C C is equivalent to the same problem for a nonexpansive and asymptotically regular mapping. This implies that fixed point results for asymptoti cally regular mappings can be applied to study the stability under renorming of the fixed point property for nonexpansive mappings. Indeed, if T : C -+ C is a nonexpansive and asymptotically regular mapping and | • | is an equivalent norm to || • | | , then the nonexpansiveness of T is not preserved when C is considered as a subset of (X, | • |), but T is still asymptotically regular on (X, | • |).
